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[GMAT math practice question]

(Geometry) Two lines \(PA\) and \(PB\) are tangent lines to the circle. What is the measure of \(∠x\)?



1) \(∠AOB = 108^o.\)

2) \(∠APB = 72^o.\)

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the value of \(x + y.\)

Follow the second and the third step: From the original condition, we have \(2\) variables (\(x\) and \(y\)). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 1 equation each, C would most likely be the answer.

Recall 3 Principles and choose C as the most likely answer. Let’s look at both conditions (1) and (2) together.

Conditions (1) and (2) tell us that \(x\) and \(y\) are positive integers and \(\frac{1}{x}+\frac{1}{y}=\frac{1}{5}\), from which we get \(\frac{1}{x}+\frac{1}{y}=\frac{1}{5}, \frac{y}{xy} + \frac{x}{xy} = \frac{1}{5}, \frac{y + x}{xy}=\frac{1}{5}, 5y + 5x = xy, xy – 5x – 5y = 0\), or \(xy – 5x – 5y + 25 = 25.\) We can factor \(xy – 5x – 5y + 25 = 25\) as follows: \((xy – 5x) + (-5y + 25) = 25, x(y – 5) + (-5)(y – 5) = 25\), which is equal to \((x - 5)(y - 5) = 25.\) We have \(3\) possible cases: \(x – 5 = 25\), and \(y – 5 = 1 / x – 5 = 5\), and \(y – 5 = 5 / x – 5 = 1\), and \(y – 5 = 25,\) since \(x\) and \(y\) are positive integers from condition (1).

Thus, the possible pairs of \(x\) and \(y\) are \(x = 6\), and \(y = \frac{30 }{ x} = 10\), and \(y = 10\) / and \(x = 30\), and \(y = 6.\)

If \(x = 6\) and \(y = 30\), then \(x + y = 36.\)

If \(x = 10\) and \(y = 10\), then \(x + y = 20.\)

The answer is not unique, and both conditions (1) and (2) combined are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions (1) and (2) together are not sufficient.
Therefore, E is the correct answer.
Answer: E

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in Common Mistake Types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question.
As shown in the figure below, if we know the measure of \(∠AOB\), we can find the measure of \(∠x\), since \(∠x = \frac{∠AOB}{2}.\)

Let’s look at each condition separately.

Condition (1) tells us that \(∠AOB = 108^o\), from which, since \(∠x = \frac{∠AOB}{2}\), we get \(∠x = \frac{∠AOB}{2}, ∠x = \frac{108^o}{2} = 54^o. \)

The answer is unique, yes, so the condition is sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.



Condition (2) tells us that \(∠APB = 72^o\). Then, since \(∠APB + ∠AOB = 180^o\), if we substitute \(∠APB = 72^o\) into this equation, we get \(72^o + ∠AOB = 180^o\), or \(∠AOB = 180^o - 72^o = 108^o,\) which is equal to condition (1). So, as shown above, this condition is also sufficient.

Also, the value of condition (1) is equal to the value of condition (2), so by Tip 1, we get D as the most likely answer.

Each condition alone is sufficient.
Therefore, D is the correct answer.
Answer: D

[GMAT math practice question]

(Ratio) At an entrance examination, the ratio of successful male applicants to successful female applicants is \(5:2\). What is the total number of applicants?

1) The ratio of male applicants to female applicants is \(3:2\).

2) The number of successful applicants is \(140\).

(Function) f(x) is a function, mapping positive integers to positive integers. What is the value of f(2) + f(3) + f(5)?

1) f(1) = 1.
2) f(a+b) = f(a) + f(b) + ab.

(Number Properties) p, q, and r are positive integers. What is the value of p + q + r?

1) p, q, and r are prime numbers.
2) The product of p, q, and r is 5 times the sum of p, q, and r.

Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

Let x and y be the number of male and female applicants, respectively. Let 5k be the number of successful male applicants and 2k be the number of successful female applicants, giving us x = 5k. Then we have to find the total number of applicants, which is equal to x + y.

Follow the second and the third step: From the original condition, we have 3 variables (x, y, and k). To match the number of variables with the number of equations, we need 3 equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer. Let’s look at both conditions 1) & 2) together.

We know that the ratio of successful male applicants to successful female applicants is 5:2; there are 100 successful male applicants and 40 successful female applicants.

Then the number of male applicants is greater than or equal to 100, and the number of female applicants is greater than or equal to 40.

If the number of male applicants is 120 and that of female applicants is 80, then the total number of applicants is 200.
If the number of male applicants is 150 and that of female applicants is 100, then the total number of applicants is 250.

The answer is not unique, and both conditions 1) and 2) together are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions (1) and (2) together are not sufficient.

Therefore, E is the correct answer.

Answer E


In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.

(Geometry) The figure shows triangle ABC and points D, E, and F, which are on line AB, BC, and CA, respectively. The area of triangle ABC is 15. What is the area of triangle BCF?



1) (BE):(EC)=3:4
2) The area of △BCF is equal to the area of □ECFD.

Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the value of f(2) + f(3) + f(5).

Follow the second and the third step: From the original condition, we have many variables to determine a function f(x). To match the number of variables with the number of equations, we need many equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer.

Let’s look at both conditions 1) & 2) together.

Since f(1) = 1, we have f(2) = f(1+1) = f(1) + f(1) + 1·1 = 1 + 1 + 1 = 3 using condition 2).
Then we have f(3) = f(2+1) = f(2) + f(1) + 2·1 = 3 + 1 + 2 = 6.

f(5) = f(3+2) = f(3) + f(2) + 3·2 = 6 + 3 + 6 = 15.

Thus, we have f(2) + f(3) + f(5) = 3 + 6 + 15 = 24.

The answer is unique, so both conditions together are sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct answer.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.

(Geometry) What is the value of ∠x + ∠y?




1) ∠BAC = \(40^o\).
2) ∠ABD = ∠DBE = ∠EBC.

Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to determine the value of p + q + r.

Follow the second and the third step: From the original condition, we have 3 variables (p, q, and r). To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer.

Let’s look at both conditions 1) & 2) together.

Since we have 3 variables (p, q, and r) and 0 equations, E is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2) together give us that:

Since we have pqr = 5(p + q + r), we assume p = 5.

Then we have 5qr = 5(5 + q + r) or qr = q + r + 5.

Since we have qr – q – r = 5, we have qr – q – r + 1 = 6 (by adding 1 to both sides so that it can easily be factored). Factoring gives us (q - 1)(r - 1) = 6.

Then we have the possible pairs of q - 1 and r - 1 are (1, 6), (2, 3), (3, 2), and (6, 1).

If we have q – 1 = 1 and r – 1 = 6, we have q = 2 and r = 7.
If we have q – 1 = 2 and r – 1 = 3, we have q = 3 and r = 4.
If we have q – 1 = 3 and r – 1 = 2, we have q = 4 and r = 3.
If we have q – 1 = 6 and r – 1 = 1, we have q = 7 and r = 2.

However, p, q, and r are prime numbers from condition 1), and their possible numbers are 2, 5, and 7.

Thus, we have p + q + r = 2 + 5 + 7 = 14.

The answer is unique, so both conditions together are sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one. So, C seems to be the answer.

However, since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately.

Condition 1) tells us that we don’t have a unique solution, obviously.

If p = 2, q = 3, and r = 5, we have p + q + r = 2 + 3 + 5 = 10.
If p = 2, q = 5, and r = 7, we have p + q + r = 2 + 5 + 7 = 14.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at the condition 2). It tells us that we don’t have a unique solution.

Assume p = 5 and since we have qr – q – r = 5, we have qr – q – r + 1 = 6 or (q - 1)(r - 1) = 6.

Then the possible pairs of q - 1 and r - 1 are (1, 6), (2, 3), (3, 2), and (6, 1).

If we have q – 1 = 1 and r – 1 = 6, we have q = 2, r = 7 and p + q + r = 5 + 2 + 7 = 14.
If we have q – 1 = 2 and r – 1 = 3, we have q = 3, r = 4 and p + q + r = 5 + 3 + 4 = 12.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Thus, really, both conditions 1) & 2) together are sufficient.

Therefore, C is the correct answer.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.

Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the area of triangle BCF, and since the area of triangle ABC is 15, we should know the ratio of AF: FC to determine the area of triangle BCF.

Follow the second and the third step: From the original condition, we have many variables (many triangles and each triangle has 3 variables). To match the number of variables with the number of equations, we need many equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer.
Let’s look at both conditions 1) & 2) together.



Since we have △BCF = △BEF + △ECF, □ECFD = △DEF + △ECF and △BCF = □ECFD, we have △BEF = △DEF.

Since triangles BEF and DEF have a common base of EF, their heights are equal to each other. Thus, AB and EF are parallel to each other.

Then triangles CEF and CBA are similar.

Since CF:AC = EC:BC = 4:7, we have △BCF = (\(\frac{4}{7}\)) △ABC = (\(\frac{4}{7}\)) * 15 = \(\frac{60}{7}\).

The answer is unique, so both conditions are sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct answer.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.

Solution:


Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the value of ∠x + ∠y.

Follow the second and the third step: From the original condition, we have many variables (many triangles and each triangle has 3 variables). To match the number of variables with the number of equations, we need many equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.
Recall 3 Principles and choose E as the most likely answer. Let’s look at both conditions 1) & 2) together.

If ∠ABD = ∠DBE = ∠EBC = \(20^o\), ∠x = \(40^o\), and ∠y = \(20^o\), we have ∠x + ∠y = \(60^o\)°.

If ∠ABD = ∠DBE = ∠EBC = \(15^o\), ∠x = \(30^o\), ∠y = \(15^o\), we have ∠x + ∠y = \(45^o\).

The answer is not unique, and the conditions combined are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are not sufficient.

Therefore, E is the correct answer.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D. 

Q1. (Numbers) N is a 3-digit positive integer. a is the hundreds digit, b the tens digit, and c the units digit. What is the maximum possible value of N?

1) b > 2a + c.
2) c > 0.

Q. (Number) What is the value of a positive integer n?

1) 756n + 576 is a perfect square number.
2) n is a unit digit number.

Q. (Number) If x and y are positive integers, what is the value of \(\frac{x}{(x + y)}\)?

1) \(\frac{y}{x}\) = \(\frac{(y - 39)}{(x - 21)}\).

2) The least common multiple of x and y is 1001.

Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

Let N = 11a + 10b + c. Then we have to find the maximum possible value of N.

Since N is a three-digit integer, we need the value of a to be the maximum possible value, and b > c.

Follow the second and the third step: From the original condition. We have 4 variables (N, a, b, and c) and 1 equation (N = 100a + 10b + c). To match the number of variables with the number of equations, we need 3 more equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer. Let’s look at both conditions (1) and (2) together. They tell us that b > 2a + c and c > 0.

The maximum value b can take is 9. For 9 > 2a + c, let’s find values for a and c.

=> c > 0 means the values c can have start at 1.

=> 9 > 2a + 1 – In this case a = 3.
=> 9 > 2a + 2 – In this case a = 3
=> 9 > 2a + 3 – In this case a = 2.

The largest possible value of a is 3, and the maximum value of c is then 2.

Then, N = 100*3+ 10*9 + 2 = 392. The answer is unique, and both conditions (1) and (2) combined are sufficient, according to CMT 2, which states that the number of answers must be only one. So, C seems to be the answer.

Since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately.

Condition (1) tells us that b > 2a + c, from which we get that for b to be greater than (2a + c), its value depends on a and c. We have digits from 0 to 9, so b cannot be greater than 9. Suppose b = 9. That means a can be 3, and c can be 2. Then we get 2a + c = 2*3 + 2 = 8 since 9 is greater than 8.

However, if a = 0, then b > c and c can have any value from 0 to 8.

Similarly, if c is 0, then b > 2a and a can have values from 0 to 4.

The answer is not unique, and the condition is not sufficient, according to CMT 2, which states that the number of answers must be only one.

Condition (2) tells us that c > 0, from which we cannot determine anything about the value of a.

The answer is not unique, and the condition is not sufficient, according to CMT 2, which states that the number of answers must be only one.

So, really, both conditions (1) and (2) combined are sufficient.

Both conditions (1) and (2) together are sufficient.

Therefore, C is the correct answer.

Answer: C